Analysis of three-dimensional extrusion of sections through curved dies by conformal transformation

A new method of analysis is proposed for the extrusion of arbitrarily shaped sections through curved die profiles. A kinematically admissible velocity field is found by deriving the equation of a stream line. Conformal transformation of a unit circle onto a section is utilized in the derivation. The upper-bound method is then applied to determine the extrusion pressure for the rigid-perfectly plastic material. The redundant work relating to the velocity discontinuities at the entrance and the exit is included in the formulation. The general formulation for an arbitrary cross section is obtained by use of conformal transformation. The upper-bound pressure for extrusion through curved die profiles is computed for a complex section with a curved boundary. Two curved die profiles widely used are chosen to compare the effects of die profiles. From the derived velocity field, the upper-bound extrusion pressures are also computed for the extrusion of regular polygons and rectangles of various aspect ratios. The effects of sectional shape, die profile and interfacial friction at the die surface are discussed.     

The use of generalised deformation boundaries for the analysis of axisymmetric extrusion through curved dies

A generalised kinematically admissible velocity field is derived for axisymmetric extrusion through curved dies by employing rigid-plastic boundaries expressed in terms of arbitrarily chosen continuous functions. The corresponding upper-bound extrusion pressure is related directly to boundary functions for the plastically deforming region when the die shape, lubrication condition and material characteristics of the billet are given. The proposed method of analysis makes it possible to predict the deformation pattern as well as extrusion pressure. In computation a third-order polynomial is chosen for the die boundary and the bounding function for the plastic region is chosen to be a fourth-order polynomial. The workhardening effect is considered in the formulation. The plastic boundaries as well as stream lines are affected by various process parameters. The theory predicts the relatively faster axial flow at the center than near the die boundary for greater friction factor even with the same die shape. The effects of area reduction and die length are also discussed in relation to extrusion pressure and deformation. Experiments are carried out for steel billets at room temperature. Deformation patterns are measured for several area reductions by the photoetching technique and the extrusion pressure is measured using a load-cell. The predicted extrusion pressure is in excellent agreement with the value computed by the finite element method. The deformation patterns agree well with the experimental observation.     

Analysis of hydrofilm extrusion of elliptic shapes using perturbation method

An approximate method of solution is proposed for the hydrofilm extrusion of elliptic shapes from round billets through optimized curved dies. A modified upper-bound theorem and hydrodynamic lubrication theory are used in combination, in order to analyze metal deformation and fluid flow respectively. The fluid analysis in hydrodynamic lubrication theory is simplified by use of elliptic transformation and perturbation technique. Strain-hardening effect of billet material and viscosity variation of fluid due to pressure are taken into consideration.For several reductions of area, experiments are carried out at room temperature by using mild steel specimens and caster oil as the lubricant.The experimental extrusion pressures are in good agreement with the theoretical predictions.     

Analysis of hydrofilm extrusion of three-dimensional shapes from round billets

A new method of die construction is proposed, which enables the exact geometrical control of die shape and ensures the initial sealing between billet and die. Using the modified method of die construction the internal metal deformation is analyzed with the aid of a special transformation combined with the upper-bound method. Based on a concept of equivalent friction factor the energy dissipation in fluid film is calculated approximately for three-dimensional hydrofilm extrusion. Computations are carried out for hydrofilm extrusion of various sections such as squares, rectangles, ellipses and clover shapes and various factors affecting the process are discussed. Experiments are performed for clover and square sections using the dies from NC manufacture based on the suggested theoretical design. A reliable sealing system is developed between punch and container. The experimental results are compared with the theoretical prediction and it is shown that the theory is in reasonable agreement with the experimental observation. Various aspects are discussed from the experiment.     

On the equivalent friction factor in hydrofilm extrusion

Hydrofilm extrusion is a kind of hydrostatic extrusion which uses a minimum amount of oil as lubricant and pressure-transmitting medium. In hydrofilm extrusion energy dissipation in the fluid lubricant between the die and working material corresponds to sliding friction in ordinary lubricated extrusion using solid lubricants. Utilizing the upper-bound theorm, an “equivalent” friction factor is defined so that the overall frictional effect between the die and working material can be conveniently investigated in terms of geometrical parameters and press velocity. On the basis of this definition, the effects of various process parameters on the frictional characteristics in hydrofilm extrusion are discussed. It is consequently found that the dominant contribution to frictional energy dissipation is made by reduction of area and press velocity. Die length is found to have very little influence on the equivalent friction factor in so far as it is longer than billet diameter.     

A study on radial flow field for extrusion through conical dies

A generalized expression for the radial flow field for extrusion through a conical die is suggested. The upper bound to the extrusion pressure for a rigid-perfectly plastic material is obtained. Other energy methods which include solutions for work-hardening and composite billets are also obtained for the generalized radial flow field. The results are used to analyse the hydrostatic extrusion of Al, Cu and Al-Cu composite billets. The extrusion pressure and hardness distribution of the product were measured in experiments and they are compared with theoretical results.     

Forward extrusion through steadily rotating conical dies. Part I: experiments

Forward extrusion of a cylindrical rod from a round billet was carried out through steadily rotating conical dies. Die rotation was shown to decrease the extrusion load and impose a twist shear strain into the partially extruded billet. The material twisting occurred both inside the container and inside the convergent die. However, not all the rotary work was transferred into shearing the bulk of the material, and this led to circumferential slippage at the rotating tool/material interface. The twisting degree on the outer surface of the material is quantified by a simple measurement technique. The influences of various process parameters on the extrusion load are also studied, including die semi-cone angle, die rotating speed and lubrication condition.     

Slip-line fields of indirect type for end extrusion through partly rough square dies

New slip-line field solutions are proposed for end extrusion through square dies with slipping friction at the workpiece-die interfaces. The fields are of indirect type and are analysed by the matrix method. The variation of the mean steady state extrusion pressure with reduction has been computed for these fields. It is shown that some of the relations between field parameters applicable to frictionless extrusion also apply to the present solutions.     

Forward extrusion through steadily rotating conical dies. Part II: theoretical analysis

Forward extrusion of a cylindrical rod from a round billet was carried out through steadily rotating conical dies. Die rotation was shown to decrease the extrusion load and impose a twist shear strain into the partially extruded billet. The material twisting occurred both inside the container and inside the convergent die. However, not all the rotary work was transferred into shearing the bulk of the material, and this led to circumferential slippage at the rotating tool/material interface. The twisting degree on the outer surface of the material is quantified by a simple measurement technique. The influences of various process parameters on the extrusion load are also studied, including die semi-cone angle, die rotating speed and lubrication condition.     

Berstorff公司的几种橡胶挤出机头

当前橡胶挤出机头类型繁多,下面重点介绍德国Berstorff公司生产的几种橡胶挤出机头： 1．异型挤出机头（如用于挤出软管的机头） 该异型机头包括机头壳体、口型板及机头锁紧装置。     

Slip-line fields for steady-state extrusion through frictionless doubly wedge-shaped dies in plane strain

Slip-line field solutions for steady extrusion through doubly wedge-shaped frictionless die are obtained by assuming a quadrilateral rigid region lying adjacent to the smaller angle wedge surface of the die. Several patterns of the slip-line field are presented. The ranges of the extrusion geometries within which those fields are valid are shown.     

A generalized spherical velocity field for bi-metallic tube extrusion through dies of any shape

In this paper, an upper-bound approach is used to analyze the extrusion process of bi-metallic tubes through dies of any shape with moving cylindrical shaped mandrel. A generalized kinematically admissible velocity field using spherical coordinate system is developed to evaluate the internal power and the power dissipated on frictional, velocity discontinuity surfaces and the total power. The extrusion process is also simulated using the finite element code, ABAQUS. Analytical results are compared with the results given by experiments of other researchers and also by the finite element method. These comparisons show a good agreement.     

支持挤出平模头自动化研磨的测量技术

针对目前国内尚无快速、经济、有效的解决挤出平模头自动化研磨平模头直线度检测的现状,通过对测量对象的深入研究,建立了测量数学模型,提出了挤出平模头直线度测量方法。该方法通过计算机对一条假设的正弦波曲线进行测量精度模拟验证,并考虑定位误差等影响最终结果的误差;对得到的目标曲线按照形位公差国家标准,使用最小条件的原则进行直线度评定。通过几组实验的数据总结,从而验证此测量方法的可行性。     

Upper bound analysis of extrusion from square billets through circular and square/rectangular dies

Upper bound elemental technique (UBET) for prediction of extrusion pressure in three-dimensional forward extrusion process is presented. Using square/rectangular billets, the study of the effect of die land length has been extended for the evaluations of extrusion pressures to extrude sections such as circular, square and rectangular shaped sections with power of deformation due to ironing effect at the die land taken into account. The extrusion pressure contributions due to the die land evaluated theoretically for these shaped sections considered are found to increase with die land lengths for any given percentage reduction and also increase with increasing percentage die reductions at any given die land length. The effect of die land lengths on the extrusion pressures increases with increasing complexity of die openings geometry with rectangular section giving the highest extrusion pressure followed by circular with square section die opening, giving the least extrusion pressure for any given die reduction at any given die land lengths. The proper choice of die land length is imperative if excessive pressure buildup at the emergent section is to be avoided so as to maintain good quality and metallurgical structure of the extrudates. This paper was recommended for publication in revised form by Associate Editor Youngseog Lee Ajiboye, Joseph S. received his B.Eng, M.Eng, and PhD degrees in Mechanical Engineering from the University of Ilorin, Nigeria, in 1988, 1995 and 2006 res-pectively. Dr. Ajiboye is a lecturer in the Department of Mechanical Engineering, Uni-versity of Lagos, Nigeria. He is currently a Contract Research Scientist at KAIST Valufacture Institute of Mechanical Engineering, School of Mechanical, Aerospace & Systems Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305 - 701, Korea. Dr. Ajiboye’s research interests include ECAE/P, determination of frictional effects in metal forming operations, upper bound and finite element in plasticity.     

Upper bound analysis of extrusion of I-section bars from square/rectangular billets through square dies

In the present study the SERR technique has been reformulated to deal with extrusion of sections with/without re-entrant corners. A comprehensive computational model has been developed for this reformulated technique. The model has been used to analyse extrusion of I-section bars and the computed results have been compared with experimental values available in literature.     

Some slip line field results for the plane strain extrusion of anisotropic materials through frictionless wedge-shaped dies

Slip line field results for the pressure to secure the plane strain extrusion of anisotropic material through frictionless wedge-shaped dies of low reduction are presented.     

Deep-drawing through tractrix type dies

The investigation evaluates the effect of die profile on the drawing performance. Three die profiles, one near to the tractrix shape and the other two with larger radius of curvature, are considered. The materials tested include mild steel, stainless steel and soft aluminium. The drawing process is analysed using a numerical solution formulated earlier and the theoretical results on the punch load and the strain development are compared with the experimental results. Good correlation is obtained on the development of strains. The theoretical prediction of the punch load is reasonably good except for mild steel which is highly anisotropic. It is shown that by modifying the tractrix die profile, the punch load can be significantly reduced.     

Service life estimation of extrusion dies by numerical simulation of fatigue-crack-growth

In this study, fatigue behavior of cold extrusion dies is investigated analytically. Experimental studies and practical experience have shown that fatigue cracks occur at the inlet radius of die shoulders. The highest stress concentration is found also at the same location. For this reason, effective stress intensity factors at different locations of the die-shoulder inlet of a typical axisymmetric extrusion die have been calculated by using the finite element method. The crack growth has been simulated by applying the Paris/Erdoĝan fatigue law to the computed data. Finally, service life of the extrusion die has been estimated from the crack-growth-rate. Agreement of theoretical estimations with available data is found to be satisfactory within the limits of experimental uncertainties. Furthermore the experimental behavior of the crack growth (stable-unstable-stable growth with final fracture) is simulated correctly. It has been shown that the unexplained behavior is caused by decrease of stress concentration with increasing crack-length.     

Laboratory assessment of wear on nitrided surfaces of dies for hot extrusion of aluminium

A newly designed “block-on-cylinder” wear resistance testing rig, which allows testing at higher contact pressures than conventional testing methods, was used to elucidate the effect of an iron nitride compound layer (white layer) on the wear resistance of nitrided dies used for hot extrusion of aluminium (Al). The tested dies (AISI H13) with various nitrided microstructures were provided by different manufacturers of equipment for plasma and gas nitriding. The wear surfaces were analysed by SEM, BEC, micrography and roughness measurements. It was found that the iron nitride compound layer is chemically more stable against hot Al in comparison to die material. Deterioration of the compound layer begins with cracking, and as a consequence, its spalling from the nitrided surface. A high thickness of the compound layer combined with a low nitriding depth leads to its earlier spalling and vice versa. Due to the increased roughness at the removal sites, accelerated chemical attack by hot Al takes place. Comparative and simultaneous testing of two nitrided samples was carried out.     

Numerical and experimental study on the three-dimensional extrusion of square section from square billet through a polynomial shaped curved die

The geometry of die profile plays a major role in reducing the extrusion pressure and ensuring the smooth flow of material. In general, the extrusion process is mostly affected by billet geometry, die geometry, and interface frictional force at the die billet geometry. In the present investigation, an analysis using three-dimensional upper bound method using fifth-order die profile function has been carried out for extrusion of square sections from square billet. The extrusion pressure and optimum die length have been computed by multivariable optimization technique. The present die shape profile is found to be superior to many other profiles. The results obtained will help in design of optimum die profile and investigation of its performance.